Is there any reference about solvability theory of quasilinear elliptic equations on closed manifolds?

In particular, I am looking for solvability condition for function $f$ of following equation $\Delta u+ c |\nabla u|^2=f$ on a closed manifold $M$, $c$ is a positive constant. As far as I understand $\int_M\, \Delta u+c |\nabla u|^2\, dV=\int_M\, c |\nabla u|^2\, dV\geq 0$, so at least $\int_M\, f\, dV\geq 0$, is that enough? Is there any sufficient condition for solvability?

Thanks a lot.

Now seems like the statement of the problem can be simplified as following

Under what condition for $f$, there exists positive solution $u$ for the Schrodinger equation $\Delta u+f u=0$ on a closed manifold.